abstract:

We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the more general class of non-trapping compact Riemannian surfaces with no conjugate points, we show lens rigidity. We also prove the injectivity of the $X$-ray transform on tensors in a variety of settings with non-convex boundary and, in some situations, allowing a non-empty trapped set.

中文翻译：

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非凸型歧管的边界和透镜刚度

摘要：

我们在不假设边界为凸面的情况下研究了边界和透镜刚度的问题。我们表明，当域是没有共轭点的简单连接的紧凑黎曼曲面时，这样的刚度成立。对于没有共轭点的非捕获紧致黎曼曲面的更一般类，我们显示了透镜的刚度。我们还证明了$ X $射线变换在具有非凸边界的各种设置中对张量的可注射性，并且在某些情况下允许使用非空的陷印集。